Stone Skipping Black Holes in Ultralight Dark Matter Solitons

TL;DR

This work identifies dipole excitations of a ULDM soliton as the key mechanism behind stone skipping—non-monotonic, quasi-periodic orbital variations of a black hole within a soliton. By combining an eigenmode decomposition of the ULDM field with simulations and a semi-analytic forced-damped oscillator model, the authors show that dipole-driven driving forces, balanced by dynamical friction, can produce resonance when the forcing frequencies approach the epicyclic frequency. The results imply that soliton backreaction can modify inspiral timescales for BHs smaller than the soliton mass and have potential consequences for SMBH dynamics and gravitational-wave observables in ULDM cosmologies. The framework also clarifies why equal-mass binaries are immune to stone skipping unless externally seeded, and it points to future work in more realistic galactic environments and multi-component ULDM models.

Abstract

The orbit of a black hole moving within an ultralight dark matter (ULDM) soliton is naively expected to decay due to dynamical friction. However, single black holes can undergo ``stone skipping'', with their orbital radius varying quasi-periodically. We show that stone skipping is induced by the dipole excitation of the soliton. We model it as resonance in a forced, damped harmonic oscillator, demonstrating that the coherent response of the soliton can significantly modify the dynamics of objects orbiting within it. This suggests that a dipole perturbation of a soliton can modify inspiral timescales if the black holes masses are significantly less than the soliton mass, with implications for supermassive black hole dynamics, the final parsec problem and gravitational wave observations in a ULDM cosmology.

Figures (10)

• Figure 1: Time evolution of the orbital radius of black holes in initially circular orbits and masses $1$, $1.5$, $2$, $3$, $4$ and $5$% of the soliton mass, from top to bottom at $t=0$. The initial orbital radii differ because, while the initial separation between the black hole and the soliton centroid is fixed at $140\,\mathrm{pc}$, more massive black holes lie closer to the system's center of mass. The physical radius of the simulation box is $1.25\,\mathrm{kpc}$.
• Figure 2: ULDM radial eigenfunctions with $n\le 4$ and $\ell\le 2$. The quantum number $n$ sets the number of radial nodes, while $\ell$ determines the small-$r$ asymptotic behavior, $f_{n\ell}(r)\propto r^\ell$ as $r\to 0$. Modes with the same $\ell$ are shown using the same color ($\ell=0$ in red, $\ell=1$ in green, and $\ell=2$ in purple); the same convention is adopted in subsequent figures.
• Figure 3: Evolution of the orbital radii of an equal-mass black hole binary, black hole masses are $2\%$ of the soliton mass.
• Figure 4: The of $|c_{n\ell m}(t)|^2$ obtained from the eigenmode decomposition of the ULDM wave functions. Top row: pure soliton without a black hole. Middle row: soliton with an equal-mass binary with each black hole mass $2\%$ of the soliton mass; no stone skipping observed. Bottom row: single black hole with mass $2\%$ of the soliton mass; stone skipping observed. Colors denote mode groups: red for the monopole, green for the dipole, and purple for the quadrupole; within each group, shading lightens as $|m|$ increases. Line styles distinguish the azimuthal order $m$: thick solid curves for $m=0$, dashed for $m>0$, and dash-dotted for $m<0$, with line width decreasing as $|m|$ increases.
• Figure 5: Evolution of the dipole component $|c_{1,1,-1}(t)|^2$ in the stone skipping run perturbed by a single black hole. The curve is well described by a $\cos^2$-type oscillation.